First the text in John 18 (NIV) seems to make clear that Caiaphas was high priest:

18:13 They bound him and brought him first to Annas, who was the father-in-law of Caiaphas, the high priest that year.

But later we read:

18:19 Meanwhile, the high priest questioned Jesus about his disciples and his teaching.
...

18:24 Then Annas sent him bound to Caiaphas the high priest.

From verses 19 and 24 it seems like Annas was the high priest, but 24 also makes clear that Caiaphas was high priest. Was there more than one high priest at a time? Or was Caiaphas somehow the one questioning Jesus in verse 19? Who was the high priest?

Annas was the official High Priest beginning in AD 6. Mosaic Law specified that the appointment of a High Priest was for life, and so he was still High Priest by Jewish standards during the time of Jesus' crucifixion.

In AD 15 Valerius Gratus (Pilate's predecessor) deposed Annas and his nephew Caiaphas was appointed. The Jews resented this arbitrary deposition -- especially by a Roman, and Annas remained High Priest in their sight. So by the time of Jesus' trial, Annas was technically (by Jewish standards) High Priest, as was his nephew Caiaphas (by both Jewish and Roman standards.)

Five of Annas' sons and his son-in-law Caiaphas served as High Priest at one time or another.1 Since Annas was the patriarch of a High-Priestly family (of which Caiaphas was a part), it made sense to bring Jesus to him first.

Note the following other passages from Scripture which lend further support to this answer:

Now in the fifteenth year of the reign of Tiberius Caesar, when Pontius Pilate was governor of Judea, and Herod was tetrarch of Galilee, and his brother Philip was tetrarch of the region of Ituraea and Trachonitis, and Lysanias was tetrarch of Abilene, in the high priesthood of Annas and Caiaphas, the word of God came to John, the son of Zacharias, in the wilderness. -Luke 3:1-2

On the next day, their rulers and elders and scribes were gathered together in Jerusalem; and Annas the high priest was there, and Caiaphas and John and Alexander, and all who were of high-priestly descent.-Acts 4:5-6

Wow, fascinating. Does Carson say how we know about Gratus deposing Annas? (Now I wonder which of them did the Yom Kippur liturgy, done by the high priest, if there were two... not this question, but your answer makes me wonder.)
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Gone QuietJun 12 '13 at 1:21

@GoneQuiet He does not. If you were interested in digging deeper, I'd recommend the Intro to the New Testament that he wrote with Moo. I'm betting that if he cites his source anywhere, it would be there.
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Jas 3.1Jun 12 '13 at 2:39

Annas had been elected high priest by the Jews, but subsequently deposed by the Romans, who appointed various successors before settling on his son-in-law, Caiaphas (Josephus, Antiquities, book XVIII 2.2). Not only was Annas highly regarded, but the Jews resented Roman interference in their religious affairs, so they still regarded him as the unofficial high priest. Thus, we have the official high priest, Caiaphas, with whom the Romans dealt, and the unofficial high priest, Annas.

Hannan (Annas) was a high priest at one point. The Romans removed him from power after his rule from 6-15AD. Because this was done by gentiles, it had almost no bearing on the Jewish community. One of their leaders would later revolt against the Romans (Shimon bar Kokhba) and one of his followers, R. Akiva, is still revered as a sage in Judaism, being a composer of the Talmud. Akiva also believed that b. Kokhba was to be the messiah, but he was later killed. We also know that, from the instructions of the torah, that there are many high priests, and that they would cast lots to see who would be in charge of the sacrifices at the Holy of Holies (Luke 1:9, also in Talmud Yoma 37a).

The high priestly function was controlled by the Romans during the time of Christ. The whole period is written up in great detail by a Galilean Jew, a Pharisee, who was a young general in the early stages of the Roman-Jewish War of AD 66-72 in which Jerusalem and the Temple were destroyed as Jesus predicted. This man's name was Joseph, or Josephus in the Latin form.

He fled from a loss in battle to a cave where he and other officers entered a suicide pact. When they had all killed themselves, he gave himself a pardon and surrendered to the Romans, becoming an advisor to General Vespasian, whom he boldly predicted would become emperor. (And he did.) In later years, Josephus felt the sting of rejection by his own people and wrote a massive volumes on The Antiquities of the Jews which sheds light on the entire Old Testament and Intertestamental period, and another titled The Wars of the Jews which tells the story of the destruction of the Temple in AD 70.

He gives us a detailed list of the high priestly family, mentions Jesus in a controversial passage, John the Baptist extensively, and the brother of Christ, "James the Just" and how his murder came about. Looking up Josephus and his works will give you intense extra-biblical background to the time of Christ.